Finite Strain Shape Memory Alloys Modeling - Scuola di Dottorato in ...

transforms according to a= Qa. Velocity is an example of a non-objective vectorbecause differentiating the rotated mapping φ = Qφwith respect to time gives,v ∂ φ ∂φ = = + φ∂tQ ∂tQ (3.76)Obviously, the magnitudes of v and v are not equal as a result of the presence of theterm Q φ , which violates the objectivity criteria.3.3.6.1 Objective stress ratesAssuming that the Cauchy stress tensor is objective, its material time derivativedefined as⎡ ∂ ⎤1σ = ( σφ) φ −(3.77)⎢ ⎣∂t⎥ ⎦is not objective. In fact, applying the chain rule, it follows that:⎡∂σ⎛ ⎞ φ⎤φ ∂σ∂σ φ φt x ⎟⎢ ⎜⎣∂ ⎝∂ ⎠ ∂t⎦⎥⎡∂σ−1⎤= +∇σ( V φ)⎢⎣∂t⎥⎦−1= + =(3.78)It can be concluded that:⎡∂σ⎤σ = + v⋅∇σ(3.79)⎢⎣∂t⎥⎦49